Halve each avocado lengthwise. Peel and pit the halves. Lay the avocados on a clean work surface, hollow-side-up. Lay a piece of plastic wrap on top, and gently press down to slightly flatten the pieces to make a flat bottom. (Keep the hollow part of the avocado intact to crack the eggs into later.) Remove plastic wrap and discard. Season avocados with salt and pepper.

Add 2 tablespoons of olive oil to a high-sided skillet and place over medium-high heat. Once hot, add the avocado slices with the hollow side facing up. Allow the avocados to sear for 1 minute. Crack 1 egg into each hole. Season the eggs with salt and pepper. Pour water into the bottom of the pan and cover the pan with a lid. Bring the water to a simmer and let the eggs steam until the egg whites have set and the yolks are still runny, about 3 minutes.

Meanwhile, in a medium bowl, mix together the tomatoes, basil, lime zest and juice, and remaining 2 tablespoons of olive oil. Season with salt and pepper.

Remove the egg-stuffed avocados to a serving platter. Top with tomato basil salad and serve.